Olympiad problems

2000 Belarus Team Selection Test, 6.2

A positive integer $A_k...A_1A_0$ is called monotonic if $A_k \le ..\le A_1 \le A_0$. Show that for any $n \in N$ there is a monotonic perfect square with $n$ digits.

1949-56 Chisinau City MO, 4

Tags: algebra , consecutive , Perfect Square

Prove that the product of four consecutive integers plus $1$ is a perfect square.

2015 Silk Road, 2

Tags: Perfect Square , Arithmetic Progression , number theory , Sequences , Sequence

Let $\left\{ {{a}_{n}} \right\}_{n \geq 1}$ and $\left\{ {{b}_{n}} \right\}_{n \geq 1}$ be two infinite arithmetic progressions, each of which the first term and the difference are mutually prime natural numbers. It is known that for any natural $n$, at least one of the numbers $\left( a_n^2+a_{n+1}^2 \right)\left( b_n^2+b_{n+1}^2 \right) $ or $\left( a_n^2+b_n^2 \right) \left( a_{n+1}^2+b_{n+1}^2 \right)$ is an perfect square. Prove that ${{a}_{n}}={{b}_{n}}$, for any natural $n$ .

2010 Hanoi Open Mathematics Competitions, 8

Tags: number theory , Perfect Squares , Perfect Square

If $n$ and $n^3+2n^2+2n+4$ are both perfect squares, find $n$.

2008 Singapore Senior Math Olympiad, 2

Tags: number theory , Perfect Square

Determine all primes $p$ such that $5^p + 4 p^4$ is a perfect square, i.e., the square of an integer.

2019 New Zealand MO, 5

Tags: number theory , Perfect Square

Find all positive integers $n$ such that $n^4 - n^3 + 3n^2 + 5$ is a perfect square.

2006 Estonia National Olympiad, 2

Tags: number theory , Perfect Square , divisible

Let $a, b$ and $c$ be positive integers such that $ab + 1, bc + 1$ and $ca + 1$ are all integer squares. a) Give an example of such numbers $a, b$ and $c$. b) Prove that at least one of the numbers $a, b$ and $c$ is divisible by $4$

2011 District Olympiad, 3

Tags: number theory , Perfect Squares , Perfect Square

A positive integer $N$ has the digits $1, 2, 3, 4, 5, 6$ and $7$, so that each digit $i$, $i \in \{1, 2, 3, 4, 5, 6, 7\}$ occurs $4i$ times in the decimal representation of $N$. Prove that $N$ is not a perfect square.

1996 Denmark MO - Mohr Contest, 4

Tags: number theory , Perfect Square , last digits , Digits

Regarding a natural number $n$, it is stated that the number $n^2$ has $7$ as the second to last digit. What is the last digit of $n^2$?

1987 Greece National Olympiad, 2

Tags: number theory , Perfect Squares , Perfect Square

Prove that exprssion $A=\frac{25}{2}(n+2-\sqrt{2n+3})$, $(n\in\mathbb{N})$ is a perfect square of an integer if exprssion $A$ is an integer .

2006 Spain Mathematical Olympiad, 2

Tags: number theory , Product , consecutive , Perfect Square , perfect cube

Prove that the product of four consecutive natural numbers can not be neither square nor perfect cube.

2010 Dutch IMO TST, 3

Tags: number theory , Perfect Square

(a) Let $a$ and $b$ be positive integers such that $M(a, b) = a - \frac1b +b(b + \frac3a)$ is an integer. Prove that $M(a,b)$ is a square. (b) Find nonzero integers $a$ and $b$ such that $M(a,b)$ is a positive integer, but not a square.

2019 Durer Math Competition Finals, 2

Tags: Perfect Square , number theory

Anne multiplies each two-digit number by $588$ in turn, and writes down the so-obtained products. How many perfect squares does she write down?

2023 SG Originals, Q4

Tags: ICMC , college contests , number theory , sum of divisors , Perfect Square

Do there exist infinitely many positive integers $m$ such that the sum of the positive divisors of $m$ (including $m$ itself) is a perfect square? [i]Proposed by Dylan Toh[/i]

1984 IMO Shortlist, 10

Tags: number theory , equation , Product , Perfect Square , IMO Shortlist

Prove that the product of five consecutive positive integers cannot be the square of an integer.

2022 3rd Memorial "Aleksandar Blazhevski-Cane", P5

Tags: algebra , Binary , Perfect Square , memorable

We say that a positive integer $n$ is [i]memorable[/i] if it has a binary representation with strictly more $1$'s than $0$'s (for example $25$ is memorable because $25=(11001)_{2}$ has more $1$'s than $0$'s). Are there infinitely many memorable perfect squares? [i]Proposed by Nikola Velov[/i]

2007 JBMO Shortlist, 3

Tags: Perfect Square , JBMO , number theory

Let $n > 1$ be a positive integer and $p$ a prime number such that $n | (p - 1) $and $p | (n^6 - 1)$. Prove that at least one of the numbers $p- n$ and $p + n$ is a perfect square.

2017 Argentina National Olympiad, 4

Tags: Perfect Square , perfect cube , number theory

For a positive integer $n$ we denote $D_2(n)$ to the number of divisors of $n$ which are perfect squares and $D_3(n)$ to the number of divisors of $n$ which are perfect cubes. Prove that there exists such that $D_2(n)=999D_3(n).$ Note. The perfect squares are $1^2,2^2,3^2,4^2,…$ , the perfect cubes are $1^3,2^3,3^3,4^3,…$ .

2020 Taiwan TST Round 2, 2

Tags: ceiling function , number theory , IMO Shortlist , IMO Shortlist 2019 , Perfect Square

Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

2019 Saudi Arabia JBMO TST, 4

Tags: number theory , Perfect Square

Let $p$ be a prime number. Show that $7^p+3p-4$ is not a perfect square.

2018 Saudi Arabia IMO TST, 2

Tags: number theory , combinatorics , Perfect Square

Let $n$ be an even positive integer. We fill in a number on each cell of a rectangle table of $n$ columns and multiple rows as following: i. Each row is assigned to some positive integer $a$ and its cells are filled by $0$ or $a$ (in any order); ii. The sum of all numbers in each row is $n$. Note that we cannot add any more row to the table such that the conditions (i) and (ii) still hold. Prove that if the number of $0$’s on the table is odd then the maximum odd number on the table is a perfect square.

2018 India PRMO, 15

Tags: minimum , number theory , Perfect Squares , Perfect Square

Let $a$ and $b$ be natural numbers such that $2a-b$, $a-2b$ and $a+b$ are all distinct squares. What is the smallest possible value of $b$ ?

2013 Saudi Arabia BMO TST, 5

Tags: number theory , Sum of Squares , Digits , Perfect Square

We call a positive integer [i]good[/i ] if it doesn’t have a zero digit and the sum of the squares of its digits is a perfect square. For example, $122$ and $34$ are good and $304$ and $12$ are not not good. Prove that there exists a $n$-digit good number for every positive integer $n$.

1974 Dutch Mathematical Olympiad, 4

Tags: Perfect Square , number theory

For which $n$ is $n^4+6n^3+11n^2+3n+31$ a perfect square?

2012 Bundeswettbewerb Mathematik, 2

Tags: number theory , Perfect Squares , Perfect Square

Are there positive integers $a$ and $b$ such that both $a^2 + 4b$ and $b^2 + 4a$ are perfect squares?